In this paper, we confirm several conjectures of Sun on Hankel-type determinants for some combinatorial sequences including Franel numbers, Domb numbers and Apéry numbers. For any nonnegative integer n, define fn:=∑k=0nnk3,Dn:=∑k=0nnk22kk2(n−k)n−k,bn:=∑k=0nnk2n+kk,An:=∑k=0nnk2n+kk2. For n=0,1,2,…, we show that 6−n|fi+j|0≤i,j≤n and 12−n|Di+j|0≤i,j≤n are positive odd integers, and 10−n|bi+j|0≤i,j≤n and 24−n|Ai+j|0≤i,j≤n are always integers. [ABSTRACT FROM AUTHOR]
Abyzov, Adel N., Danchev, Peter V., and Tapkin, Daniel T.
Subjects
JACOBSON radical, ALGEBRA, MATHEMATICS, INTEGERS
Abstract
Let R be a ring and let n be an arbitrary but fixed positive integer. We characterize those rings R whose elements a satisfy at least one of the relations that a n + a or a n − a is a nilpotent whenever n ∈ ℕ ∖ { 1 }. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with x n − x nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with x n − x nilpotent, J. Algebra Appl. 21 (2022)], respectively. [ABSTRACT FROM AUTHOR]